Question: The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=\frac{n+1}{2n+5}$. Write a rule for ${{a}_{n}}$.
Explanation: An individual term of a series is the difference between two successive partial sums. In particular, ${{a}_{n}}={{S}_{n}}-{{S}_{n-1}}\,$. $\begin{aligned} {{S}_{n}}-{{S}_{n-1}}&=\frac{n+1}{2n+5}-\frac{n}{2n+3} \\\\ &=\frac{(n+1)(2n+3)-n(2n+5)}{(2n+5)(2n+3)} \\\\ &=\frac{3}{(2n+5)\left( 2n+3 \right)} \end{aligned}$